Goursat tetrahedron

From Wikipedia, the free encyclopedia
Jump to: navigation, search
For Euclidean 3-space, there are 3 simple and related Goursat tetrahedra, represented by [4,3,4], [4,31,1], and [3[4]]. They can be seen inside as points on and within a cube, {4,3}.

In geometry, a Goursat tetrahedron is a tetrahedral fundamental domain of a Wythoff construction. Each tetrahedral face represents a reflection hyperplane on 3-dimensional surfaces: the 3-sphere, the Euclidean 3-space, and hyperbolic 3-space. Coxeter named them after Édouard Goursat who first looked into these domains. It is an extension of the theory of Schwarz triangles for Wythoff constructions on the sphere.

Graphical representation[edit]

A Goursat tetrahedron can be represented graphically by a tetrahedral graph, which is in a dual configuration of the fundamental domain tetrahedron. In the graph, each node represents a face (mirror) of the Goursat tetrahedron. Each edge is labeled by a rational value corresponding to the reflection order, being π/dihedral angle.

General Goursat tetrahedron.png

A 4-node Coxeter-Dynkin diagram represents this tetrahedral graphs with order-2 edges hidden. If many edges are order 2, the Coxeter group can be represented by a bracket notation.

Existence requires each of the 3-node subgraphs of this graph, (p q r), (p u s), (q t u), and (r s t), must correspond to a Schwarz triangle.

Extended symmetry[edit]

Tetrahedral subgroup tree.png Tetrahedron symmetry tree.png
The symmetry of a Goursat tetrahedron can be tetrahedral symmetry of any subgroup symmetry shown in this tree, with subgroups below with subgroup indices labeled in the colored edges.

An extended symmetry of the Goursat tetrahedron is a semidirect product of the Coxeter group symmetry and the fundamental domain symmetry (the Goursat tetrahedron in these cases). Coxeter notation supports this symmetry as double-brackets like [Y[X]] means full Coxeter group symmetry [X], with Y as a symmetry of the Goursat tetrahedron. If Y is a pure reflective symmetry, the group will represent another Coxeter group of mirrors. If there is only one simple doubling symmetry, Y can be implicit like [[X]] with either reflectional or rotational symmetry depending on the context.

The extended symmetry of each Goursat tetrahedron is also given below. The highest possible symmetry is that of the regular tetrahedron as [3,3], and this occurs in the prismatic point group [2,2,2] or [2[3,3]] and the paracompact hyperbolic group [3[3,3]].

See Tetrahedron#Isometries_of_irregular_tetrahedra for 7 lower symmetry isometries of the tetrahedron.

Whole number solutions[edit]

The following sections show all of the whole number Goursat tetrahedral solutions on the 3-sphere, Euclidean 3-space, and Hyperbolic 3-space. The extended symmetry of each tetrahedron is also given.

The colored tetrahedal diagrams below are vertex figures for omnitruncated polytopes and honeycombs from each symmetry family. The edge labels represent polygonal face orders, which is double the Coxeter graph branch order. The dihedral angle of an edge labeled 2n is π/n. Yellow edges labeled 4 come from right angle (unconnected) mirror nodes in the Coxeter diagram.

3-sphere (finite) solutions[edit]

Finite Coxeter groups isomorphisms

The solutions for the 3-sphere with density 1 solutions are: (Uniform polychora)

Duoprisms and hyperprisms:
Coxeter group
and diagram
[2,2,2]
CDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.png
[p,2,2]
CDel node.pngCDel p.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.png
[p,2,q]
CDel node.pngCDel p.pngCDel node.pngCDel 2.pngCDel node.pngCDel q.pngCDel node.png
[p,2,p]
CDel node.pngCDel p.pngCDel node.pngCDel 2.pngCDel node.pngCDel p.pngCDel node.png
[3,3,2]
CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.png
[4,3,2]
CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.png
[5,3,2]
CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.png
Group symmetry order 16 8p 4pq 4p2 48 96 240
Tetrahedron
symmetry
[3,3]
(order 24)
Regular tetrahedron diagram.png
[2]
(order 4)
Digonal disphenoid diagram.png
[2]
(order 4)
Digonal disphenoid diagram.png
[2+,4]
(order 8)
Tetragonal disphenoid diagram.png
[ ]
(order 2)
Sphenoid diagram.png
[ ]+
(order 1)
Scalene tetrahedron diagram.png
[ ]+
(order 1)
Scalene tetrahedron diagram.png
Extended symmetry [(3,3)[2,2,2]]
CDel node c1.pngCDel 2.pngCDel node c1.pngCDel 2.pngCDel node c1.pngCDel 2.pngCDel node c1.png
=[4,3,3]
CDel node c1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
[2[p,2,2]]
CDel node c1.pngCDel p.pngCDel node c1.pngCDel 2.pngCDel node c2.pngCDel 2.pngCDel node c2.png
=[2p,2,4]
CDel node.pngCDel 2x.pngCDel p.pngCDel node c1.pngCDel 2.pngCDel node.pngCDel 4.pngCDel node c2.png
[2[p,2,q]]
CDel node c1.pngCDel p.pngCDel node c1.pngCDel 2.pngCDel node c2.pngCDel q.pngCDel node c2.png
=[2p,2,2q]
CDel node.pngCDel 2x.pngCDel p.pngCDel node c1.pngCDel 2.pngCDel node.pngCDel 2x.pngCDel q.pngCDel node c2.png
[(2+,4)[p,2,p]]
CDel node c1.pngCDel p.pngCDel node c1.pngCDel 2.pngCDel node c1.pngCDel p.pngCDel node c1.png
=[2+[2p,2,2p]]
CDel node.pngCDel 2x.pngCDel p.pngCDel node c1.pngCDel 2.pngCDel node c1.pngCDel 2x.pngCDel p.pngCDel node.png
[1[3,3,2]]
CDel node c1.pngCDel 3.pngCDel node c2.pngCDel 3.pngCDel node c1.pngCDel 2.pngCDel node c3.png
=[4,3,2]
CDel node.pngCDel 4.pngCDel node c1.pngCDel 3.pngCDel node c2.pngCDel 2.pngCDel node c3.png
[4,3,2]
CDel node c1.pngCDel 4.pngCDel node c2.pngCDel 3.pngCDel node c3.pngCDel 2.pngCDel node c4.png
[5,3,2]
CDel node c1.pngCDel 5.pngCDel node c2.pngCDel 3.pngCDel node c3.pngCDel 2.pngCDel node c4.png
Extended symmetry order 384 32p 16pq 32p2 96 96 240
Graph type Linear Tridental
Coxeter group
and diagram
Pentachoric
[3,3,3]
CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
Hexadecachoric
[4,3,3]
CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
Icositetrachoric
[3,4,3]
CDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
Hexacosichoric
[5,3,3]
CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
Demitesseractic
[31,1,1]
CDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.png
Vertex figure of omnitruncated uniform polychora
Tetrahedron Omnitruncated 5-cell verf.png Omnitruncated 8-cell verf.png Omnitruncated 24-cell verf.png Omnitruncated 120-cell verf.png Omnitruncated demitesseract verf.png
Group symmetry order 120 384 1152 14400 192
Tetrahedron
symmetry
[2]+
(order 2)
Half-turn tetrahedron diagram.png
[ ]+
(order 1)
Scalene tetrahedron diagram.png
[2]+
(order 2)
Half-turn tetrahedron diagram.png
[ ]+
(order 1)
Scalene tetrahedron diagram.png
[3]
(order 6)
Isosceles trigonal pyramid diagram.png
Extended symmetry [2+[3,3,3]]
CDel branch c1.pngCDel 3ab.pngCDel nodeab c2.png
[4,3,3]
CDel node c1.pngCDel 4.pngCDel node c2.pngCDel 3.pngCDel node c3.pngCDel 3.pngCDel node c4.png
[2+[3,4,3]]
CDel label4.pngCDel branch c1.pngCDel 3ab.pngCDel nodeab c2.png
[5,3,3]
CDel node c1.pngCDel 5.pngCDel node c2.pngCDel 3.pngCDel node c3.pngCDel 3.pngCDel node c4.png
[3[31,1,1]]
CDel nodeab c1.pngCDel split2.pngCDel node c2.pngCDel 3.pngCDel node c1.png
=[3,4,3]
CDel node c2.pngCDel 3.pngCDel node c1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
Extended symmetry order 240 384 2304 14400 1152

Euclidean (affine) 3-space solutions[edit]

Euclidean Coxeter group isomorphisms

Density 1 solutions: Convex uniform honeycombs:

Graph type Linear Tri-dental Loop Prismatic Degenerate
Coxeter group
Coxeter diagram
[4,3,4]
CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
[4,31,1]
CDel nodes.pngCDel split2.pngCDel node.pngCDel 4.pngCDel node.png
[3[4]]
CDel branch.pngCDel 3ab.pngCDel branch.png
[4,4,2]
CDel node.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node.png
[6,3,2]
CDel node.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.png
[3[3],2]
CDel branch.pngCDel split2.pngCDel node.pngCDel 2.pngCDel node.png
[∞,2,∞]
CDel node.pngCDel infin.pngCDel node.pngCDel 2.pngCDel node.pngCDel infin.pngCDel node.png
Vertex figure of omnitruncated honeycombs
Tetrahedron Omnitruncated cubic honeycomb verf.png Omnitruncated alternated cubic honeycomb verf.png Omnitruncated 3-simplex honeycomb verf.png
Tetrahedron
Symmetry
[2]+
(order 2)
Half-turn tetrahedron diagram.png
[ ]
(order 2)
Sphenoid diagram.png
[2+,4]
(order 8)
Tetragonal disphenoid diagram.png
[ ]
(order 2)
Sphenoid diagram.png
[ ]+
(order 1)
Scalene tetrahedron diagram.png
[3]
(order 6)
Isosceles trigonal pyramid diagram.png
[2+,4]
(order 8)
Tetragonal disphenoid diagram.png
Extended symmetry [(2+)[4,3,4]]
CDel branch c2.pngCdel 4-4.pngCDel nodeab c1.png
[1[4,31,1]]
CDel nodeab c1.pngCDel split2.pngCDel node c2.pngCDel 4.pngCDel node c3.png
=[4,3,4]
CDel node.pngCDel 4.pngCDel node c1.pngCDel 3.pngCDel node c2.pngCDel 4.pngCDel node c3.png
[(2+,4)[3[4]]]
CDel branch c1.pngCDel 3ab.pngCDel branch c1.png
=[2+[4,3,4]]
CDel branch c1.pngCdel 4-4.pngCDel nodes.png
[1[4,4,2]]
CDel node c1.pngCDel 4.pngCDel node c2.pngCDel 4.pngCDel node c1.pngCDel 2.pngCDel node c3.png
=[4,4,2]
CDel node.pngCDel 4.pngCDel node c1.pngCDel 4.pngCDel node c2.pngCDel 2.pngCDel node c3.png
[6,3,2]
CDel node c1.pngCDel 6.pngCDel node c2.pngCDel 3.pngCDel node c3.pngCDel 2.pngCDel node c4.png
[3[3[3],2]]
CDel branch c1.pngCDel split2.pngCDel node c1.pngCDel 2.pngCDel node c2.png
=[3,6,2]
CDel node.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node c1.pngCDel 2.pngCDel node c2.png
[(2+,4)[∞,2,∞]]
CDel node c1.pngCDel infin.pngCDel node c1.pngCDel 2.pngCDel node c1.pngCDel infin.pngCDel node c1.png
=[1[4,4]]
CDel node.pngCDel 4.pngCDel node c1.pngCDel 4.pngCDel node.png

Compact hyperbolic 3-space solutions[edit]

Density 1 solutions: (Convex uniform honeycombs in hyperbolic space) (Coxeter_diagram#Compact (Lannér simplex groups))

Rank 4 Lannér simplex groups
Graph type Linear Tri-dental
Coxeter group
Coxeter diagram
[3,5,3]
CDel node.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png
[5,3,4]
CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
[5,3,5]
CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png
[5,31,1]
CDel node.pngCDel 5.pngCDel node.pngCDel split1.pngCDel nodes.png
Vertex figures of omnitruncated honeycombs
Tetrahedron Omnitruncated icosahedral honeycomb verf.png Omnitruncated order-4 dodecahedral honeycomb verf.png Omnitruncated order-5 dodecahedral honeycomb verf.png Omnitruncated alternated order-5 cubic honeycomb verf.png
Tetrahedron
Symmetry
[2]+
(order 2)
Half-turn tetrahedron diagram.png
[ ]+
(order 1)
Scalene tetrahedron diagram.png
[2]+
(order 2)
Half-turn tetrahedron diagram.png
[ ]
(order 2)
Sphenoid diagram.png
Extended symmetry [2+[3,5,3]]
CDel label5.pngCDel branch c1.pngCDel 3ab.pngCDel nodeab c2.png
[5,3,4]
CDel node c1.pngCDel 5.pngCDel node c2.pngCDel 3.pngCDel node c3.pngCDel 4.pngCDel node c4.png
[2+[5,3,5]]
CDel branch c1.pngCDel 5a5b.pngCDel nodeab c2.png
[1[5,31,1]]
CDel node c1.pngCDel 5.pngCDel node c2.pngCDel split1.pngCDel nodeab c3.png
=[5,3,4]
CDel node c1.pngCDel 5.pngCDel node c2.pngCDel 3.pngCDel node c3.pngCDel 4.pngCDel node.png
Graph type Loop
Coxeter group
Coxeter diagram
[(4,3,3,3)]
CDel label4.pngCDel branch.pngCDel 3ab.pngCDel branch.png
[(4,3)2]
CDel label4.pngCDel branch.pngCDel 3ab.pngCDel branch.pngCDel label4.png
[(5,3,3,3)]
CDel label5.pngCDel branch.pngCDel 3ab.pngCDel branch.png
[(5,3,4,3)]
CDel label5.pngCDel branch.pngCDel 3ab.pngCDel branch.pngCDel label4.png
[(5,3)2]
CDel label5.pngCDel branch.pngCDel 3ab.pngCDel branch.pngCDel label5.png
Vertex figures of omnitruncated honeycombs
Tetrahedron Uniform t0123 4333 honeycomb verf.png Uniform t0123 4343 honeycomb verf.png Uniform t0123 5333 honeycomb verf.png Uniform t0123 5343 honeycomb verf.png Uniform t0123 5353 honeycomb verf.png
Tetrahedron
Symmetry
[2]+
(order 2)
Half-turn tetrahedron diagram.png
[2,2]+
(order 4)
Rhombic disphenoid diagram.png
[2]+
(order 2)
Half-turn tetrahedron diagram.png
[2]+
(order 2)
Half-turn tetrahedron diagram.png
[2,2]+
(order 4)
Rhombic disphenoid diagram.png
Extended symmetry [2+[(4,3,3,3)]]
CDel label4.pngCDel branch c1.pngCDel 3ab.pngCDel branch c2.png
[(2,2)+[(4,3)2]]
CDel label4.pngCDel branch c1.pngCDel 3ab.pngCDel branch c1.pngCDel label4.png
[2+[(5,3,3,3)]]
CDel label5.pngCDel branch c1.pngCDel 3ab.pngCDel branch c2.png
[2+[(5,3,4,3)]]
CDel label5.pngCDel branch c1.pngCDel 3ab.pngCDel branch c2.pngCDel label4.png
[(2,2)+[(5,3)2]]
CDel label5.pngCDel branch c1.pngCDel 3ab.pngCDel branch c1.pngCDel label5.png

Paracompact hyperbolic 3-space solutions[edit]

This show subgroup relations of paracompact hyperbolic Goursat tetrahedra. Order 2 subgroups represent bisecting a Goursat tetrahedron with a plane of mirror symmetry
Hyperbolic subgroup tree 344.png

Density 1 solutions: (See Coxeter diagram#Paracompact (Koszul simplex groups))

Rank 4 Koszul simplex groups
Graph type Linear graphs
Coxeter group
and diagram
[6,3,3]
CDel node.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
[3,6,3]
CDel node.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.png
[6,3,4]
CDel node.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
[6,3,5]
CDel node.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png
[6,3,6]
CDel node.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node.png
[4,4,3]
CDel node.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
[4,4,4]
CDel node.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.png
Tetrahedron
symmetry
[ ]+
(order 1)
Scalene tetrahedron diagram.png
[2]+
(order 2)
Digonal disphenoid diagram.png
[ ]+
(order 1)
Scalene tetrahedron diagram.png
[ ]+
(order 1)
Scalene tetrahedron diagram.png
[2]+
(order 2)
Digonal disphenoid diagram.png
[ ]+
(order 1)
Scalene tetrahedron diagram.png
[2]+
(order 2)
Digonal disphenoid diagram.png
Extended symmetry [6,3,3]
CDel node c1.pngCDel 6.pngCDel node c2.pngCDel 3.pngCDel node c3.pngCDel 3.pngCDel node c4.png
[2+[3,6,3]]
CDel label6.pngCDel branch c1.pngCDel 3ab.pngCDel nodeab c2.png
[6,3,4]
CDel node c1.pngCDel 6.pngCDel node c2.pngCDel 3.pngCDel node c3.pngCDel 4.pngCDel node c4.png
[6,3,5]
CDel node c1.pngCDel 6.pngCDel node c2.pngCDel 3.pngCDel node c3.pngCDel 5.pngCDel node c4.png
[2+[6,3,6]]
CDel branch c1.pngCDel 6a6b.pngCDel nodeab c2.png
[4,4,3]
CDel node c1.pngCDel 4.pngCDel node c2.pngCDel 4.pngCDel node c3.pngCDel 3.pngCDel node c4.png
[2+[4,4,4]]
CDel label4.pngCDel branch c1.pngCdel 4-4.pngCDel nodeab c2.png
Graph type Loop graphs
Coxeter group
and diagram
[3[ ]×[ ]]
CDel node.pngCDel split1.pngCDel branch.pngCDel split2.pngCDel node.png
[(4,4,3,3)]
CDel node.pngCDel split1-44.pngCDel nodes.pngCDel split2.pngCDel node.png
[(43,3)]
CDel label4.pngCDel branch.pngCdel 4-4.pngCDel branch.png
[4[4]]
CDel label4.pngCDel branch.pngCdel 4-4.pngCDel branch.pngCDel label4.png
[(6,33)]
CDel label6.pngCDel branch.pngCDel 3ab.pngCDel branch.pngCDel 2.png
[(6,3,4,3)]
CDel label6.pngCDel branch.pngCDel 3ab.pngCDel branch.pngCDel label4.png
[(6,3,5,3)]
CDel label6.pngCDel branch.pngCDel 3ab.pngCDel branch.pngCDel label5.png
[(6,3)[2]]
CDel label6.pngCDel branch.pngCDel 3ab.pngCDel branch.pngCDel label6.png
Tetrahedron
symmetry
[2]
(order 4)
Digonal disphenoid diagram.png
[ ]
(order 2)
Sphenoid diagram.png
[2]+
(order 2)
Digonal disphenoid diagram.png
[2+,4]
(order 8)
Tetragonal disphenoid diagram.png
[2]+
(order 2)
Digonal disphenoid diagram.png
[2]+
(order 2)
Digonal disphenoid diagram.png
[2]+
(order 2)
Digonal disphenoid diagram.png
[2,2]+
(order 4)
Tetragonal disphenoid diagram.png
Extended symmetry [2[3[ ]×[ ]]]
CDel node c2.pngCDel split1.pngCDel branch c1.pngCDel split2.pngCDel node c2.png
=[6,3,4]
CDel node.pngCDel 6.pngCDel node c1.pngCDel 3.pngCDel node c2.pngCDel 4.pngCDel node.png
[1[(4,4,3,3)]]
CDel node c1.pngCDel split1-44.pngCDel nodeab c3.pngCDel split2.pngCDel node c2.png
=[3,41,1]
CDel node.pngCDel 4.pngCDel node c3.pngCDel split1-43.pngCDel nodeab c1-2.png
[2+[(43,3)]]
CDel label4.pngCDel branch c1.pngCdel 4-4.pngCDel branch c2.png
[(2+,4)[4[4]]]
CDel label4.pngCDel branch c1.pngCdel 4-4.pngCDel branch c1.pngCDel label4.png
=[2+[4,4,4]]
CDel label4.pngCDel branch c1.pngCdel 4-4.pngCDel nodes.png
[2+[(6,33)]]
CDel label6.pngCDel branch c1.pngCDel 3ab.pngCDel branch c2.pngCDel 2.png
[2+[(6,3,4,3)]]
CDel label6.pngCDel branch c1.pngCDel 3ab.pngCDel branch c2.pngCDel label4.png
[2+[(6,3,5,3)]]
CDel label6.pngCDel branch c1.pngCDel 3ab.pngCDel branch c2.pngCDel label5.png
[(2,2)+[(6,3)[2]]]
CDel label6.pngCDel branch c1.pngCDel 3ab.pngCDel branch c1.pngCDel label6.png
Graph type Tri-dental Loop-n-tail Simplex
Coxeter group
and diagram
[6,31,1]
CDel node.pngCDel 6.pngCDel node.pngCDel split1.pngCDel nodes.png
[3,41,1]
CDel node.pngCDel 3.pngCDel node.pngCDel split1-44.pngCDel nodes.png
[41,1,1]
CDel node.pngCDel 4.pngCDel node.pngCDel split1-44.pngCDel nodes.png
[3,3[3]]
CDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel branch.png
[4,3[3]]
CDel node.pngCDel 4.pngCDel node.pngCDel split1.pngCDel branch.png
[5,3[3]]
CDel node.pngCDel 5.pngCDel node.pngCDel split1.pngCDel branch.png
[6,3[3]]
CDel node.pngCDel 6.pngCDel node.pngCDel split1.pngCDel branch.png
[3[3,3]]
CDel branch.pngCDel splitcross.pngCDel branch.png
Tetrahedron
symmetry
[ ]
(order 2)
Sphenoid diagram.png
[ ]
(order 2)
Sphenoid diagram.png
[3]
(order 6)
Isosceles trigonal pyramid diagram.png
[ ]
(order 2)
Sphenoid diagram.png
[ ]
(order 2)
Sphenoid diagram.png
[ ]
(order 2)
Sphenoid diagram.png
[ ]
(order 2)
Sphenoid diagram.png
[3,3]
(order 24)
Regular tetrahedron diagram.png
Extended symmetry [1[6,31,1]]
CDel node c1.pngCDel 6.pngCDel node c2.pngCDel split1.pngCDel nodeab c3.png
=[6,3,4]
CDel node c1.pngCDel 6.pngCDel node c2.pngCDel 3.pngCDel node c3.pngCDel 4.pngCDel node.png
[1[3,41,1]]
CDel node c1.pngCDel 3.pngCDel node c2.pngCDel split1-44.pngCDel nodeab c3.png
=[3,4,4]
CDel node c1.pngCDel 3.pngCDel node c2.pngCDel 4.pngCDel node c3.pngCDel 4.pngCDel node.png
[3[41,1,1]]
CDel node c1.pngCDel 4.pngCDel node c2.pngCDel split1-44.pngCDel nodeab c1.png
=[4,4,3]
CDel node c2.pngCDel 4.pngCDel node c1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
[1[3,3[3]]]
CDel node c1.pngCDel 3.pngCDel node c2.pngCDel split1.pngCDel branch c3.png
=[3,3,6]
CDel node c1.pngCDel 3.pngCDel node c2.pngCDel 3.pngCDel node c3.pngCDel 6.pngCDel node.png
[1[4,3[3]]]
CDel node c1.pngCDel 4.pngCDel node c2.pngCDel split1.pngCDel branch c3.png
=[4,3,6]
CDel node c1.pngCDel 4.pngCDel node c2.pngCDel 3.pngCDel node c3.pngCDel 6.pngCDel node.png
[1[5,3[3]]]
CDel node c1.pngCDel 5.pngCDel node c2.pngCDel split1.pngCDel branch c3.png
=[5,3,6]
CDel node c1.pngCDel 5.pngCDel node c2.pngCDel 3.pngCDel node c3.pngCDel 6.pngCDel node.png
[1[6,3[3]]]
CDel node c1.pngCDel 6.pngCDel node c2.pngCDel split1.pngCDel branch c3.png
=[6,3,6]
CDel node c1.pngCDel 6.pngCDel node c2.pngCDel 3.pngCDel node c3.pngCDel 6.pngCDel node.png
[(3,3)[3[3,3]]]
CDel branch c1.pngCDel splitcross.pngCDel branch c1.png
=[6,3,3]
CDel node c1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png

Rational solutions[edit]

There are hundreds of rational solutions for the 3-sphere, including these 6 linear graphs which generate the Schläfli-Hess polychora, and 11 nonlinear ones from Coxeter:

Linear graphs
  1. Density 4: [3,5,5/2] CDel node.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.pngCDel 5.pngCDel rat.pngCDel d2.pngCDel node.png
  2. Density 6: [5,5/2,5] CDel node.pngCDel 5.pngCDel node.pngCDel 5.pngCDel rat.pngCDel d2.pngCDel node.pngCDel 5.pngCDel node.png
  3. Density 20: [5,3,5/2] CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 5.pngCDel rat.pngCDel d2.pngCDel node.png
  4. Density 66: [5/2,5,5/2] CDel node.pngCDel 5.pngCDel rat.pngCDel d2.pngCDel node.pngCDel 5.pngCDel node.pngCDel 5.pngCDel rat.pngCDel d2.pngCDel node.png
  5. Density 76: [5,5/2,3] CDel node.pngCDel 5.pngCDel node.pngCDel 5.pngCDel rat.pngCDel d2.pngCDel node.pngCDel 3.pngCDel node.png
  6. Density 191: [3,3,5/2] CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 5.pngCDel rat.pngCDel d2.pngCDel node.png
Loop-n-tail graphs:
  1. Density 2: CDel label3-2.pngCDel branch.pngCDel split2.pngCDel node.pngCDel 5.pngCDel node.png
  2. Density 3: CDel label5.pngCDel branch.pngCDel split2-5t.pngCDel node.pngCDel 3.pngCDel node.png
  3. Density 5: CDel label5-3.pngCDel branch.pngCDel split2-53.pngCDel node.pngCDel 3.pngCDel node.png
  4. Density 8: CDel label5-4.pngCDel branch.pngCDel split2-55.pngCDel node.pngCDel 3.pngCDel node.png
  5. Density 9: CDel branch.pngCDel split2-p3.pngCDel node.pngCDel 3.pngCDel node.png
  6. Density 14: CDel label5.pngCDel branch.pngCDel split2-p3.pngCDel node.pngCDel 5.pngCDel node.png
  7. Density 26: CDel label5-3.pngCDel branch.pngCDel split2-p3.pngCDel node.pngCDel 5.pngCDel node.png
  8. Density 30: CDel branch.pngCDel split2-5p.pngCDel node.pngCDel 3.pngCDel node.png
  9. Density 39: CDel label3-2.pngCDel branch.pngCDel split2-53.pngCDel node.pngCDel 3.pngCDel node.png
  10. Density 46: CDel label5.pngCDel branch.pngCDel split2-5t.pngCDel node.pngCDel 5.pngCDel rat.pngCDel d2.pngCDel node.png
  11. Density 115: CDel label5.pngCDel branch.pngCDel split2-p3.pngCDel node.pngCDel 3.pngCDel node.png

See also[edit]

References[edit]

  • Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8 (page 280, Goursat's tetrahedra) [1]
  • Norman Johnson The Theory of Uniform Polytopes and Honeycombs, Ph.D. (1966) He proved the enumeration of the Goursat tetrahedra by Coxeter is complete
  • Goursat, Edouard, Sur les substitutions orthogonales et les divisions régulières de l'espace, Annales scientifiques de l'École Normale Supérieure, Sér. 3, 6 (1889), (pp. 9–102, pp. 80–81 tetrahedra)
  • Klitzing, Richard. "Dynkin Diagrams Goursat tetrahedra". 
  • Norman Johnson, Geometries and Transformations, Chapters 11,12,13, preprint 2011
  • N. W. Johnson, R. Kellerhals, J. G. Ratcliffe, S. T. Tschantz, The size of a hyperbolic Coxeter simplex, Transformation Groups 1999, Volume 4, Issue 4, pp 329–353 [2]